Undecidability of topological and arithmetical properties of infinitary rational relations

Finkel, Olivier

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 37 (2003) no. 2, pp. 115-126.

We prove that for every countable ordinal $\alpha$ one cannot decide whether a given infinitary rational relation is in the Borel class ${\Sigma }_{\alpha }^{\mathbf{0}}$ (respectively ${\Pi }_{\alpha }^{\mathbf{0}}$). Furthermore one cannot decide whether a given infinitary rational relation is a Borel set or a ${\Sigma }_{\mathbf{1}}^{\mathbf{1}}$-complete set. We prove some recursive analogues to these properties. In particular one cannot decide whether an infinitary rational relation is an arithmetical set. We then deduce from the proof of these results some other ones, like: one cannot decide whether the complement of an infinitary rational relation is also an infinitary rational relation.

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